Perhaps one reason for this effectiveness of mathematics is that many laws of physics can be expressed in terms of minimizing free energy or minimizing action. These optimization problems have mathematical solutions.

In general, surface energies become more important than bulk energies at small scales: A small bug can easily walk on water, because the force of surface tension outweighs gravity at that small scale.

Thus problems about real-world materials, especially those concerning structure at small scales, can often be cast in the form of optimizing some feature of shape. The system minimizes some energy depending on the shape of a surface (or sometimes a curve) describing the material's structure. Mathematically, we obtain an optimization problem for some geometric energy.

A classical example is the soap bubble which minimizes its area while enclosing a fixed volume; this leads to the study of the more general constant-mean-curvature surfaces found in bubble-clusters and foams. Biological cell membranes, on the other hand, are more complicated bilayer surfaces, and seem to minimize an elastic bending energy known as the Willmore energy.

My own mathematical research concerns geometric optimization problems like these. I have looked also at singularities in higher-dimensional soap films, configurations of points on a sphere, and ropelength of knots. Ropelength describes how to optimally tie a given type of knot in a piece of real rope, and it seems to be related to the physical behavior of knotted loops of bacterial DNA.